Microstrip Patch Antennas Introduction – Transmission Line Model

Microstrip antennas (or patch antennas) are extremely important in modern electrical engineering for the simple fact that they can directly be printed to a circuit board. This makes them necessary for things like cellular antennas for GPS, communication with cell towers and bluetooth/WiFi. Patch antennas are notoriously narrowband, especially those with a rectangular shape (patch antennas can have a wide variety of shapes). Patch antennas can be configured as single antennas or in an array. The excitation is usually fed by a microstrip line which usually has a characteristic impedance of 50 ohms.

One of the most common analysis methods for analyzing microstrip antennas is the transmission line model. It is important to note that the microstrip transmission line does not support TEM mode, unlike the coaxial cable which has radial symmetry. For the microstrip line, quasi-TEM is supported. For this mode, there is a field component along the direction of propagation, although it is small. For the purposes of the model, this can be ignored and the TEM mode which has no field component in the direction of propagation can be used. This reduces the model to:


Where the effective dielectric constant can be approximated as:


The width of the strip must be greater than the height of the substrate. It is important to note that the dielectric constant is not constant for frequency. As a consequence, the above approximation is only valid for low frequencies of microwave.

Another note for the transmission line model is that the effective length differs from the physical length of the patch. The effective length is longer by 2ΔL due to fringing effects. ΔL can be expressed as a function of the effective dielectric constant.





The Helical Antenna

The helical antenna is a frequently overlooked antenna type commonly used for VHF and UHF applications and provides high directivity, wide bandwidth and interestingly, circular polarization. Circular polarization provides a huge advantage in that if two antennas are circularly polarized, the will not suffer polarization loss due to polarization mismatch. It is known that circular polarization is a special case of elliptical polarization. Circular polarization occurs when the Electric field vector (which defines the polarization of any antenna) has two components which are in quadrature with equal amplitudes. In this case, the electric field vector rotates in a circular pattern when observed at the target, whether it be RHP or LHP (right hand or left hand polarized).

Generally, the axial mode of the helix antenna is used but normal mode may also be used. Usually the helix is mounted on a ground plane which is connected to a coaxial cable using a N type or SMA connector.

The helix antenna can be broken down into triangles, shown below.


The circumference of each loop is given by πD. S represents the spacing between loops. When this is zero (and hence the angle of the triangle is zero), the helix antenna reduces to a flat loop. When the angle becomes a 90 degree angle, the helix reduces to a monopole linear wire antenna. L0 represents the length of one loop and L is the length of the entire antenna. The total height L is given as NS, where N is the number of loops. The actual length can be calculated by multiplying the number of loops with the length of one loop L0.

An important thing to note is that the helix antenna is elliptically polarized by default and must be manually designed to achieve circular polarization for a specific bandwidth. Another note is that the input impedance of the antenna depends greatly on the pitch angle (alpha).

The axial (endfire) mode, which is more common occurs when the circumference of the antenna is roughly the size of the wavelength. This mode is easier to achieve circular polarization. The normal mode features a much smaller circumference and is more omnidirectional in terms of radiation pattern.

The Axial ratio is the numerical quantity that governs the polarization. When AR = 1, the antenna is circularly polarized. When AR = ∞ or 0, the antenna is linearly polarized. Any other quantity means elliptical polarization.


The axial ratio can also be approximated by:


For axial mode, the radiation pattern is much more directional, as the axis of the antenna contains the bulk of the radiation. For this mode, the following conditions must be met to achieve circular polarization.


These are less stringent than the normal mode conditions.

It is also important to consider that the input impedance of these antennas tends to be higher than the standard impedance of a coaxial line (100-200 ohms compared to 50). Flattening the feed wire of the antenna and covering the ground plane with dielectric material helps achieve a better SWR.


This equation can be used to calculated the height of the dielectric used for the ground plane. It is dependent on the transmission line characteristic impedance, strip width and the dielectric constant of the material used.

The Superheterodyne Receiver

“Heterodyning” is a commonly used term in the design of RF wireless communication systems. It the process of using a local oscillator of a frequency close to an input signal in order to produce a lower frequency signal on the output which is the difference in the two frequencies. It is contrasted with “homodyning” which uses the same frequency for the local oscillator and the input. In a superhet receiver, the RF input and the local oscillator are easily tunable whereas the ouput IF (intermediate frequency) is fixed.


After the antenna, the front end of the receiver comprises of a band select filter and a LNA (low noise amplifier). This is needed because the electrical output of the antenna is often as small as a few microvolts and needs to be amplified, but not in a way that leads to a higher Noise Figure. The typical superhet NF should be around 8-10 dB. Then the signal is frequency multiplied or heterodyned with the local oscillator. In the frequency domain, this corresponds to a shift in frequency. The next filter is the channel select filter which has a higher Quality factor than the band select filter for enhanced selectivity.

For the filtering, the local oscillator can either be fixed or variable for downconversion to the baseband IF. If it is variable, a variable capacitor or a tuning diode is used. The local oscillator can be higher or lower in frequency than the desired frequency resulting from the heterodyning (high side or low side injection).

A common issue in the superhet receiver is image frequency, which needs to be suppressed by the initial filter to prevent interference. Often multiple mixer stages are used (called multiple conversion) to overcome the image issue. The image frequencies are given below.


Higher IF frequencies tend to be better at suppressing image as demonstrated in the term 2f_IF. The level of attenuation (in dB) of a receiver to image is given in the Image Rejection Ratio (the ratio of the output of the receiver from a signal at the received frequency, to its output for an equal strength signal at the image frequency.





PN Junction under biasing (conduction and valence band energies):


Code Used (PN Junction):

#TOP TO BOTTOM – Structure Specification


PIN Junction Biased:


PIN Junction Unbiased:


Code Used (PIN Junction):

#TOP TO BOTTOM – Structure Specification

Here, the carrier concentrations are plotted:


RADAR Range Resolution

Before delving into the topic of pulse compression, it is necessary to briefly discuss the advantages of pulse RADAR over CW RADAR. The main difference between the two is with duty cycle (time high vs total time). For CW RADARs this is 100% and pulse RADARs are typically much lower. The efficiency of this comes with the fact that the scattered signal can be observed when the signal is low, making it much more clear. With CW RADARs (which are much less common then pulse RADARs), since the transmitter is constantly transmitting, the return signal must be read over the transmitted signal. In all cases, the return signal is weaker than the transmitter signals due to absorption by the target. This leads to difficulties with continuous wave RADAR.  Pulse RADARs can also provide high peak power without increasing average power, leading to greater efficiency.

“Pulse Compression” is a signal processing technique that tries to take the advantages of pulse RADAR and mitigate its disadvantages. The major dilemma is that accuracy of RADAR is dependent on pulse width. For instance, if you send out a short pulse you can illuminate the target with a small amount of energy. However the range resolution is increased. The digital processing of pulse compression grants the best of both worlds: having a high range resolution and also illuminate the target with greater energy. This is done using Linear Frequency Modulation or “Chirp modulation”, illustrated below.


As shown above, the frequency gradually increases with time (x axis).

A “matched filter” is a processing technique to optimize the SNR, which outputs a compressed pulse.

Range resolution can be calculated as follows:

Resolution = (C*T)/2

Where T is the pulse time or width.

With greater range resolution, a RADAR can detect two objects that are very close. As shown this is easier to do with a longer pulse, unless pulse compression is achieved.

It can also be demonstrated that range resolution is proportional to bandwidth:

Resolution = c/2B

So this means that RADARs with higher frequencies (which tend to have higher bandwidth), greater resolution can also be achieved.



Energy Bandgaps



  • GaAs: ~ 1.2 eV
  • InP: ~ 1.35 eV
  • AlGaAs: ~ 1.8 eV
  • InGaAs: ~0.75 eV
  • InGaAsP: 1.1 eV


Here the conduction band and valence band are shown.


The structure used in the PN Junction Simulator is found below:

#TOP TO BOTTOM – Structure Specification

Is the bandgap affected by doping the concentration level?





PN Junction Simulator in ATLAS


go atlas


#Define the mesh

mesh auto
x.m l = -2 Spac=0.1
x.m l = -1 Spac=0.05
x.m l = 1 Spac=0.05
x.m l = 2 Spac =0.1

#TOP TO BOTTOM – Structure Specification

#Electrode specification
#Gate Metal Work Function
contact num=2 work=4.77
models region=1 print conmob fldmob srh optr
models region=2 srh optr
material region=2

solve init outf=diode_mb1.str master
output con.band val.band
tonyplot diode_mb1.str

method newton autonr trap maxtrap=6 climit=1e-6
solve vanode = 2.5 name=anode
save outfile=diode_mb2.str
tonyplot diode_mb2.str




Here is the pn junction with no biasing.


The beam profile can also be obtained:


ATLAS TCAD: Simulation of Frequency Response from Light Impulse


Generic Light Frequency Response Simulator Program in ATLAS TCAD


go atlas

[define mesh]

[define structure]

[define electrodes]

[define materials]





solve init
outf = lightpulse_frequencyresponse.str
LOG lightpulse_frequencyresponse.log

[simulation conditions such as applied voltage]

LOG off

Now the optical pulse is is simulated as follows:

LOG outf=transient.log

tonyplot transient.log

outf=lightpulse_frequencyresponse.str master onefile
log off


log outf=frequencyplot.log
tonyplot frequencyplot.log
log off


save outf=lightpulse_frequencyresponse.str
tonyplot lightpulse_frequencyresponse.str






High Speed UTC Photodetector Simulation with Frequency Response in TCAD




I-V Curve


Beam Simulation Entering Photodetector:



Light Impulse:


Frequency Response in ATLAS:


The full project (pdf) is here: ece530_final_mbenker


Sinusoidal and Exponential Sequences, Periodicity of Sequences

Continuing our discussion on discrete-time sequences, we now come to define exponential and sinusoidal sequences. The general formula for a discrete-time exponential sequence is as follows:

x[n] = Aα^n.

This exponential behaves differently according to the value of α. If the sequence starts at n=0, the formula is as follows:

x[n] = Aα^n * u[n].


If α is a complex number, the exponential function exhibits newer characteristics. The envelope of the exponential is |α|. If |α| < 1, the system is decaying. If |α|> 1, the system is growing.


When α is complex, the sequence may be analyzed as follows, using the definition of Euler’s formula to express a complex relationship as a magnitude and phase difference.

Captu56 ma

Where ω0 is the frequency and φ is the phase, for n number of samples, a complex exponential sequence of form Ae^jw0n may be considered as a sinusoidal sequence for a set of frequencies in an interval of 2π.

A sinusoidal sequence is defined as follows:

x[n] = Acos(ω0*n + φ), for all n, and A, φ are real constants.

Periodicity for discrete-time signals means that the sequence will repeat itself for a certain delay, N.

x[n] = x[n+N] : system is periodic.

t = (-5:1:15)’;

impulse = t==0;
unitstep = t>=0;
Alpha1 = -0.5;
Alpha2 = 0.5;
Alpha3 = 2.5;
Alpha4 = -2.5;
cAlpha1 = -0.5 – 0.5i;
cAlpha2 = 0.5 + 0.5i;
cAlpha3 = 2.5 -2.5i;
cAlpha4 = -2.5 + 2.5i;
A = 1;

Exp1 = A.*unitstep.*Alpha1.^t;
Exp2 = A.*unitstep.*Alpha2.^t;
Exp3 = A.*unitstep.*Alpha3.^t;
Exp4 = A.*unitstep.*Alpha4.^t;

cExp1 = A.*unitstep.*cAlpha1.^t;
cExp2 = A.*unitstep.*cAlpha2.^t;
cExp3 = A.*unitstep.*cAlpha3.^t;
cExp4 = A.*unitstep.*cAlpha4.^t;

stem(t, impulse)

stem(t, unitstep)
title(‘Unit Step’)
stem(t, cExp1)
title(‘Exponential: alpha = -0.5 – 0.5i’)

stem(t, cExp2)
title(‘Exponential: alpha = 0.5 + 0.5i’)

stem(t, cExp3)
title(‘Exponential: alpha = 2.5 -2.5i’)

stem(t, cExp4)
title(‘Exponential: alpha = -2.5 + 2.5i’)
stem(t, Exp1)
title(‘Exponential: alpha = -0.5’)

stem(t, Exp2)
title(‘Exponential: alpha = 0.5’)

stem(t, Exp3)
title(‘Exponential: alpha = 2.5’)

stem(t, Exp4)
title(‘Exponential: alpha = -2.5’)



Mathematical Formulation for Antennas: Radiation Integrals and Auxiliary Potentials

This short paper will attempt to clarify some useful mathematical tools for antenna analysis that seem overly “mathematical” but can aid in understanding antenna theory. A solid background in Maxwell’s equations and vector calculus would be helpful.

Two sources will be introduced: The Electric and Magnetic sources (E and M respectively). These will be integrated to obtain either an electric and magnetic field directly or integrated to obtain a Vector potential, which is then differentiated to obtain the E and H fields. We will use A for magnetic vector potential and F for electric vector potential.

Using Gauss’ laws (first two equations) for a source free region:


And also the identity:


It can be shown that:


In the case of the magnetic field in response to the magnetic vector potential (A). This is done by equating the divergence of B with the divergence of the curl of A, which both equal zero. The same can be done from Gauss Law of electricity (1st equation) and the divergence of the curl of F.

Using Maxwell’s equations (not necessary to know how) the following can be derived:


For total fields, the two auxiliary potentials can be summed. In the case of the Electric field this leads to:


The following integrals can be used to solve for the vector potentials, if the current densities are known:


For some cases, the volume integral is reduced to a surface or line integral.

An important note: most antenna calculations and also the above integrals are independent of distance, and therefore are done in the far field (region greater than 2D^2/λ, where D is the largest dimension of the antenna).

The familiar duality theorem from Fourier Transform properties can be applied in a similar way to Maxwell’s equations, as shown.


In the chart, Faraday’s Law, Ampere’s Law, Helmholtz equations and the above mentioned integrals are shown. To be perfectly honest, I think the top right equation is wrong. I believe is should have permittivity rather than permeability.

Another important antenna property is reciprocity… that is the receive and transmit radiation patterns are the same , given that the medium of propagation is linear and isotropic. This can be compared to the reciprocity theorem of circuits, meaning that a volt meter and source can be interchanged if a constant current or voltage source is used and the circuit components are linear, bilateral and discrete elements.


Discrete-Time Impulse and Unit Step Functions

Discrete-Time Signals are understood as a set or sequence of numbers. These sequences possess magnitudes or values at a given index.

One mark of Discrete-Time Signals is that the index value is an integer. Thus, the sequence will have a magnitude or value for a whole number index such as -5, -4, 0, 6, 10000, etc.

A discrete-time signal represented as a sequence of numbers takes the following form:

x[n] = {x[n]},          -∞ < n < ∞,

where n is any real integer (the index).

An analog representation describes values of a signal at time nT, where T is the sampling period. The sampling frequency is the inverse of the sampling period.

x[n] = X_a(nT),      -∞ < n < ∞.


Common Sequences

Both a very simple and important sequence is the unit sample sequence, “discrete time impulse” or simply “impulse,” equal to 1 only at zero and equal to zero otherwise.


The discrete time impulse is used to describe an entire system using a delayed impulse. An entire sequence may also be shifted or delayed using the following relation:

y[n] = x[n – n0],

where n0 is an integer (which is the increment of indices by which the system is delayed. The impulse function delayed to any index and multiplied by the value of the system at that index can describe any discrete-time system. The general formula for this relationship is,


The unit step sequence is related to the unit impulse. The unit step sequence is a set of numbers that is equal to zero for all numbers less than zero and equal to one for numbers equal and greater than zero.


The unit step sequence is therefore equal to a sequence of delta impulses with a zero and greater delay.

u[n] = δ[n] + δ[n-1] + δ[n-2] + . . .


The unit impulse can also be represented by unit step functions:

δ[n] = u[n] – u[n-1].

Below I’ve plotted both the impulse and unit step function in matlab.


t = (-10:1:10)';

impulse = t==0;
unitstep = t>=0;

stem(t, impulse)
stem(t, unitstep)
title('Unit Step')



Image Resolution

Consider that we are interested in building an optical sensor. This sensor contains a number of pixels, which is dependent on the size of the sensor. The sensor has two dimensions, horizontal and vertical. Knowing the size of the pixels, we will be able to find the total number of pixels on this sensor.

The horizontal field of view, HFOV is the total angle of view normal from the sensor. The effective focal length, EFL of the sensor is then:

Effective Focal Length: EFL = V / (tan(HFOV/2)),

where V is the vertical sensor size in (in meters, not in number of pixels) and HFOV is the horizontal field of view. Horizontal field of view as an angled is halved to account that HFOV extends to both sizes of the normal of the sensor.

The system resolution using the Kell Factor: R = 1000 * KellFactor * (1 / (PixelSize)),

where the Pixel size is typically given and the Kell factor, less than 1 will approximate a best real case result and accounts for aberrations and other potential issues.

Angular resolution: AR = R * EFL / 1000,

where R is the resolution using the Kell factor and EFL is the effective focal length. It is possible to compute the angular resolution using either pixels per millimeter or cycles per millimeter, however one would need to be consistent with units.

Minimum field of view: Δl = 1.22 * f * λ / D,

which was used previously for the calculation of the spatial resolution of a microscope. The minimum field of view is exactly a different wording for the minimum spatial resolution, or minimum size resolvable.

Below is a MATLAB program that computed these parameters, while sweeping the diameter of the lens aperture. The wavelength admittedly may not be appropriate for a microscope, but let’s say that you are looking for something in the infrared spectrum. Maybe you are trying to view some tiny laser beams that will be used in the telecom industry at 1550 nanometer.

Pixel size: 3 um. HFOV: 4 degrees. Sensor size: 8.9mm x 11.84mm.


Spatial Resolution of a Microscope

Angular resolution describes the smallest angle between two objects that are able to be resolved.

θ = 1.22 * λ / D,

where λ is the wavelength of the light and D is the diameter of the lens aperture.

Spatial resolution on the other hand describes the smallest object that a lens can resolve. While angular resolution was employed for the telescope, the following formula for spatial resolution is applied to microscopes.

Spatial resolution: Δl = θf = 1.22 * f * λ / D,

where θ is the angular resolution, f is the focal length (assumed to be distance to object from lens as well), λ is the wavelength and D is the diameter of the lens aperture.



The Numerical Aperture (NA) is a measure of the the ability to of the lens to gather light and resolve fine detail. In the case of fiber optics, the numerical aperture applies to the maximum acceptance angle of light entering a fiber. The angle by the lens at its focus is θ = 2α. α is shown in the first diagram.

Numerical Aperture for a lens: NA = n * sin(α),

where n is the index of refraction of the medium between the lens and the object. Further,

sin(α) = D / (2d).

The resolving power of a microscope is related.

Resolving power: x = 1.22 * d * λ / D,

where d is the distance from the lens aperture to the region of focus.


Using the definition of NA,

Resolving power: x = 1.22 * d * λ / D = 1.22 * λ / (2sin(α)) = 0.61 * λ / NA.


Telescope Resolution & Distance Between Stars using the Rayleigh Limit

Previously, the Rayleigh Criterion and the concept of maximum resolution was explained. As mentioned, Rayleigh found this formula performing an experiment with telescopes and stars, exploring the concept of resolution. This formula may be used to determine the distance between two stars.

θ = 1.22 * λ / D.

Consider a telescope of lens diameter of 2.4 meters for a star of visible white light at approximately 550 nanometer wavelength. The distance between the two stars in lightyears may be calculated as follows. The stars are approximately 2.6 million lightyears away from the lens.

θ = 1.22 * (550*10^(-9)m)/(2.4m)

θ =2.80*10^(-7) rad

Distance between two objects (s) at a distance away (r), separated by angle (θ): s = rθ

s = rθ = (2.0*10^(6) ly)*(2.80*10^(-7)) = 0.56 ly.

This means that the maximum resolution for the lens size, star distance from the lens and wavelength would be that two stars would need to be separated at least 0.56 lightyears for the two stars to be distinguishable.


Electrical Engineering Students at University of Massachusetts Dartmouth